Symmetric Circuits for Rank Logic

Fixed-point logic with rank (FPR) is an extension of fixed-point counting (FPC) operators for computing the a matrix over finit field. The expressive power FPR properly extends that FPC and contained in P, but it not known if containment proper. We give circuit characterization terms families symmetric circuits gates, along lines given by Anderson Dawar 2017. This requires development broad framework which individual gates compute functions are (i.e., invariant under all permutations their inputs). also necessitates novel techniques to prove equivalence logic. Both greater generality than main result.